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Axioms 2019, 8(2), 42; https://doi.org/10.3390/axioms8020042

The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis

Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia
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Current address: The Faculty of Mathematics and Mechanics of Saint Petersburg State University, 28 Universitetsky pr., 198504 St. Petersburg, Russia.
Received: 24 February 2019 / Revised: 31 March 2019 / Accepted: 4 April 2019 / Published: 10 April 2019
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Abstract

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions. View Full-Text
Keywords: axiomatic non-standard analysis; hyperrational numbers; line segment measurement axiomatic non-standard analysis; hyperrational numbers; line segment measurement
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Lovyagin, Y.N.; Lovyagin, N.Y. The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis. Axioms 2019, 8, 42.

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